An old conjecture of Erdos–Turán on additive bases

Abstract

There is a 1941 conjecture of Erdős and Turán on what is now called additive basis that we restate: Conjecture 0.1(Erdős and Turán). Suppose that 0 = δ 0 > δ 1 > δ 2 > δ 3 ⋯ 0 = \delta _0>\delta _1>\delta _2>\delta _3\cdots is an increasing sequence of integers and \[ s ( z ) := ∑ i = 0 ∞ z δ i . s(z) : = \sum _{i=0}^\infty z^{\delta _i}. \] Suppose that \[ s 2 ( z ) := ∑ i = 0 ∞ b i z i . s^2(z) := \sum _{i=0}^\infty b_i z^i. \] If b i > 0 b_i>0 for all i i , then { b n } \{b_n\} is unbounded. Our main purpose is to show that the sequence { b n } \{b_n\} cannot be bounded by 7 7 . There is a surprisingly simple, though computationally very intensive, algorithm that establishes this.